in the last chapter demonstrates that

P — —ih ( L 4- ■h cLCm L) (3.2,2)

with the domain

tf), p t p e • (3,2.3) |

ÿ In lackey's scheme the Hamiltonian is assumed to be proportional -A to the Laplacian, Mackey’s scheme also deals with the problem of

coordinate variables and their functions. Tlte result is that associated with every real-valued Borel function f defined everywhere on the

manifold M there corresponds a quantum observable which is just the familiar self-adjoint operation of m ultiplication by f in the Hilbert

space L^(M). A coordinate variable x^, i f defined throughout M,

is a Borel function on M. Hence, i t may be quantized in the usual way. The situation is different i f x^ is not defined throughout M as

would be the case when the manifold M cannot be covered by a single chart.

We are now in a position to re-examine the canonical quantization sdieme to see why i t sometimes breaks down. Let us consider a manifold M which is cover able by a single coordinate chart x^. C lassically,

a generalized momentim may be regarded as a function on T*M. As sudi, i f i t is associated with an OPG of M in the manner described above, may be satisfa ctorily quantized to give a self-adjoint quantum momentum observable P. whose restriction to C“(M) is

(3,2.4) where is the generator of the OPG, In this case we. are

ju stified in calling x^,Pj^ canonical variables. The crucial criterion is tiiat Pj^ must generate an OPG of M. Otherwise the differential operator (3,2,4) w ill not be essen tially self-adjoint, Consequently, we w ill not have a well-defined quantum momentum observable. The

;

fact tliat the generator is the vector field h/bx t e lls us that

tlie OPG generated by the generalized momentum associated with a i

generalized coordinate x is the group of translation of tlie coor-

i • i • ’

din ate x , However, translations form a group only i f x is allowed

tlie range f- oô^+oo). This is precisely why a quantm canonical momentum

observable can be established only for a coordinate variable which takes the range (- + oo).

§3, Some Applications of Mackey’s Scheme

I

§(3,1) Che-Dimensional Manifolds J

Throughout this subsection and unless indicated to the contrary

we w ill assume that the metric form of M is ds^ = dx^. Also we note |

that one way of finding an OPG of M is to find a diffeomorphism | F:M->/R, If F is so, then U (M) ^ F“^(t + F(m)) (te/R,meI^ is an OPG

J

coverable by a single chart, namely m->F(m), If M is not, then we have to appeal to another way in order to find an OPG of M.

§C3,1,1) M “ the real lin e /P

An OPG of IR is the group of translations U^(x) - x + t . The

infinitesim al generator of is L - d/dx. Since div L = 0 , the

momentum operator takes the form P = -ihd/dx. Because L is complete,

the operator P with a demain 1)^ — 6 L ( M ) }

is self-adjoint, Tlie Hamiltonian operator of a free particle is

2

- ^ = |p 2 . The domain of H is ={ tl'PeDp.P'ReDp} [2,p318].

Hie Hamiltonian with the previous domain is self-adjoint.

Let y be a coordinate system on IR in terms of which the mefric assumes the form

ds^ - g dy^. (3.3.1)

Let Ly. d/dy be the infinitesim al generator o f an OPG of |R . If we

require ly to be yolujiie-preserving [28], then

div L = ^ ('U = 0 . - (3.3.2) I

The solution of the above d ifferen tial equation is

Ç = k /(J' , (3.3.3)

assumes tlie form

= ( - i. A //s ' ) djdÿ . (3.3.4)

prom [3.3,4] the quantum HamiItalian is

If we le t fqr example , y = arctan x, then ds^ = dx^ = dy^/cos^y.

The OPG U^(y] = arctan(t + tan y) has the infinitesim al generator

2 ^

L = cos y d/dy Wiich is volume-preserving. The momentum operator P #

y y 'f

and the Hamiltonian H are given by

Py. = -ih cos^y d/dy , H = -|h^(cos^y d/dy)^.

In terms of x, the OPG is just the translation group = x + t .

/ *f*

We may perform on /R another OPG U^(x] = xe , Tlie generator

L = X d/dx of is of divergence equal to 1. The corresponding

A A

momentum operator is P = -ih(x d/dx + |] , P is self-adjoint when the

2 2

usual domain (3,2.3] is chosen. The free Haniltonian is -Jh V as usual, but i t is no longer proportional to the square of the momentum operator. We note that the classical momentum P = xp

arising fran is not a constant of the free motion and possesses

no obvious physical significance.

§(3.1.2) M = the semi-real line (0,o^)

While the translation of the coordinate is not an OPG of M, "t

the transformations U^(x) = xe form an OPG of M, The generator

L of U. , the momentum operator and the Hamiltoiian are x d/dx'

2 d2 '

-ih(x d/dx + I) and -|h —% respectively. The momentum operator dx

with the domain (3.2.3) is self-ad joint but is not compatible ifith the Hamiltonian.

§(3.1,2) M = the open interval (-^^, M 1 Since P(x) = tan x is a diJEfeomorpiiism £rcm MtoiR, we have j

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